57:020 Mechanics of Fluids and Transport Processes



Chapter 3 Bernoulli Equation3.1 Flow Patterns: Streamlines, Pathlines, StreaklinesA streamline ψx,t is a line that is everywhere tangent to the velocity vector at a given instant.Examples of streamlines around an airfoil (left) and a car (right)A pathline is the actual path traveled by a given fluid particle.An illustration of pathline (left) and an example of pathlines, motion of water induced by surface waves (right)A streakline is the locus of particles which have earlier passed through a particular point. An illustration of streakline (left) and an example of streaklines, flow past a full-sized streamlined vehicle in the GM aerodynamics laboratory wind tunnel, and 18-ft by 34-ft test section facilility by a 4000-hp, 43-ft-diameter fan (right)Note:For steady flow, all 3 coincide.For unsteady flow, ψt pattern changes with time, whereas pathlines and streaklines are generated as the passage of timeStreamline:By definition we must have V×dr=0 which upon expansion yields the equation of the streamlines for a given time t=t1dxu=dyv=dzw=dswhere s = integration parameter. So if (u, v, w) know, integrate with respect to s for t=t1 with I.C. (x0, y0, z0, t1) at s=0 and then eliminate s.Pathline:The path line is defined by integration of the relationship between velocity and displacement.dxdt=u dydt=v dzdt=wIntegrate u, v, w with respect to t using I.C. (x0, y0, z0, t0) then eliminate t.Streakline:To find the streakline, use the integrated result for the pathline retaining time as a parameter. Now, find the integration constant which causes the pathline to pass through (x0, y0, z0) for a sequence of time ξ<t. Then eliminate ξ.3.2 Streamline CoordinatesEquations of fluid mechanics can be expressed in different coordinate systems, which are chosen for convenience, e.g., application of boundary conditions: Cartesian (x, y, z) or orthogonal curvilinear (e.g., r, θ, z) or non-orthogonal curvilinear. A natural coordinate system is streamline coordinates (s, n, l); however, difficult to use since solution to flow problem (V) must be known to solve for steamlines.For streamline coordinates, since V is tangent to s there is only one velocity component.Vx,t=vsx,ts+vnx,tnwhere vn=0 by definition.Figure 4.8 Streamline coordinate system for two-dimensional flow.The acceleration isa=DVDt=?V?t+V??Vwhere,?=??ss+??nn; V??=vs??sa=ass+ann=?V?t+vs?V?s =?vs?ts+vs?s?t+vs?vs?ss+vs?s?sFigure 4.9 Relationship between the unit vector along the streamline, s, and the radius of curvature of the streamline, Rs+?s?sds?θ?sdsndθsNormal to sds=RdθSpace increments+?θ?sdsn=s+?s?sds?s?s=nRs+?s?tdt?θ?tdtndθss+?θ?tdtn=s+?s?tdt?s?t=?θ?tn are chosen for convenience, e.g.,GM aerodynTime incrementa=?vs?t+vs?vs?ss+vs?θ?t?vn?t+vs2Rnoras=?vs?t+vs?vs?s, an=?vn?t+vs2Rwhere, ?vs?t = local as in s direction?vn?t = local an in n directionvs?vs?s = convective as due to spatial gradient of V i.e. convergence /divergence ψvs2R = convective an due to curvature of : centrifugal accerleration3.3 Bernoulli EquationConsider the small fluid particle of size δs by δn in the plane of the figure and δy normal to the figure as shown in the free-body diagram below. For steady flow, the components of Newton’s second law along the streamline and normal directions can be written as following:1) Along a streamlineδm?as=∑δFs=δWs+δFpswhere,δm?as=ρδV?vs?vs?sδWs=-γδVsinθδFps=p-δpsδnδy-p+δpsδnδy=-2δpsδnδyδps=?p?sδs2 1st order Taylor Series =-?p?sδVThus,ρδV?vs?vs?s=-?p?sδV-γδVsinθsinθ=dzds3629025108585ρvs?vs?s=-?p?s-γsinθ =-??sp+γz change in speed due to ?p?s and ?z?s (i.e. W along s)2) Normal to a streamlineδm?an=∑δFn=δWn+δFpnwhere,δm?an=ρδV?vs2RδWn=-γδVcosθ δpn=?p?nδn2 1st order Taylor Series δFpn=p-δpnδsδy-p+δpnδsδy=-2δpnδsδy =-?p?nδV Thus,cosθ=dzdn3924300274955ρδV?vs2R=-?p?nδV-γδVcosθ ρvs2R=-?p?n-γcosθ =-??np+γz streamline curvature is due to ?p?n and ?z?n (i.e. W along n)In a vector form:ρa=-?p+γz (Euler equation)or ρvs?vs?ss+vs2Rn=-??ss+??nnp+γz Steady flow, ρ = constant, s equationρvs?vs?s=-??sp+γz ??svs22+pρ+gz=0 ∴ vs22+pρ+gz=constantBernoulli equationSteady flow, ρ = constant, n equationρvs2R=-??np+γz ∴ vs2Rdn+pρ+gz=constantFor curved streamlines p+γz (= constant for static fluid) decreases in the n direction, i.e. towards the local center of curvature.It should be emphasized that the Bernoulli equation is restricted to the following:inviscid flowsteady flowincompressible flowflow along a streamlineNote that if in addition to the flow being inviscid it is also irrotational, i.e. rotation of fluid = ω = vorticity = ?×V = 0, the Bernoulli constant is same for all ψ, as will be shown later.3.4 Physical interpretation of Bernoulli equationIntegration of the equation of motion to give the Bernoulli equation actually corresponds to the work-energy principle often used in the study of dynamics. This principle results from a general integration of the equations of motion for an object in a very similar to that done for the fluid particle. With certain assumptions, a statement of the work-energy principle may be written as follows:The work done on a particle by all forces acting on the particle is equal to the change of the kinetic energy of the particle.The Bernoulli equation is a mathematical statement of this principle.In fact, an alternate method of deriving the Bernoulli equation is to use the first and second laws of thermodynamics (the energy and entropy equations), rather than Newton’s second law. With the approach restrictions, the general energy equation reduces to the Bernoulli equation.An alternate but equivalent form of the Bernoulli equation ispγ+V22g+z=constantalong a streamline.Pressure head: pγVelocity head: V22gElevation head: zThe Bernoulli equation states that the sum of the pressure head, the velocity head, and the elevation head is constant along a streamline.3.5 Static, Stagnation, Dynamic, and Total Pressurep+12ρV2+γz=pT=constantalong a streamline.Static pressure: pDynamic pressure: 12ρV2Hydrostatic pressure: γzStagnation points on bodies in flowing fluids.Stagnation pressure: p+12ρV2 (assuming elevation effects are negligible)Total pressure: pT=p+12ρV2+γz (along a streamline) The Pitot-static tube (left) and typical Pitot-static tube designs (right).Typical pressure distribution along a Pitot-static tube.3.6 Applications of Bernoulli Equation1) Stagnation Tubep1+ρV122=p2+ρV222V12=2ρp2-p1=2ργlV1=2glz1=z2p1=γd, V2=0p2=γl+d gageLimited by length of tube and need for free surface reference2) Pitot Tubep1γ+V122g+z1=p2γ+V222g+z2V2=2gp1γ+z1h1-p2γ+z2h212where, V1=0 and h = piezometric headV=V2=2gh1-h2h1-h2 from manometer or pressure gageFor gas flow Δpγ?ΔzV=2Δpρ3) Free JetsVertical flow from a tankApplication of Bernoulli equation between points (1) and (2) on the streamline shown givesp1+12ρV12+γz1=p2+12ρV22+γz2Since z1=h, z2=0, V1≈0, p1=0, p2=0, we haveγh=12ρV22V2=2γhρ=2ghBernoulli equation between points (1) and (5) givesV5=2gh+H4) Simplified form of the continuity equationSteady flow into and out of a tankObtained from the following intuitive arguments:Volume flow rate: Q=VAMass flow rate: m=ρQ=ρVAConservation of mass requiresρ1V1A1=ρ2V2A2For incompressible flow ρ1=ρ2, we haveV1A1=V2A2or Q1=Q25) Volume Rate of Flow (flowrate, discharge)1. Cross-sectional area oriented normal to velocity vector (simple case where V⊥A)U = constant: Q = volume flux = UA [m/s m2 = m3/s]U≠ constant: Q=AUdASimilarly the mass flux = m=AρUdA2. General caseQ=CSV?ndA=CSVcosθdAm=CSρV?ndAAverage velocity:V=QAExample:At low velocities the flow through a long circular tube, i.e. pipe, has a parabolic velocity distribution (actually paraboloid of revolution).u=umax1-rR2where, umax = centerline velocitya) find Q and VQ=AV?ndA=AudAAudA=02π0Rurrdθdr=2π0Rurrdrwhere, dA=2πrdr, u=ur and not θ, ∴ 02πdθ=2πQ=2π0Rumax1-rR2rdr=12umaxπR2V=QA=umax26) Flowrate measurementVarious flow meters are governed by the Bernoulli and continuity equations.Typical devices for measuring flowrate in pipes.Three commonly used types of flow meters are illustrated: the orifice meter, the nozzle meter, and the Venturi meter. The operation of each is based on the same physical principles—an increase in velocity causes a decrease in pressure. The difference between them is a matter of cost, accuracy, and how closely their actual operation obeys the idealized flow assumptions.We assume the flow is horizontal (z1=z2), steady, inviscid, and incompressible between points (1) and (2). The Bernoulli equation becomes:p1+12ρV12=p2+12ρV22If we assume the velocity profiles are uniform at sections (1) and (2), the continuity equation can be written as:Q=V1A1=V2A2where A2 is the small (A2<A1) flow area at section (2). Combination of these two equations results in the following theoretical flowrateQ=A22p1-p2ρ1-A2A12assumed vena contracta = 0, i.e., no viscous effects. Otherwise,Q=CCAC2p1-p2ρ1-A2A12where CC = contraction coefficient A smooth, well-contoured nozzle (left) and a sharp corner (right)The velocity profile of the left nozzle is not uniform due to differences in elevation, but in general d?h and we can safely use the centerline velocity, V2, as a reasonable “average velocity.” For the right nozzle with a sharp corner, dj will be less than dh. This phenomenon, called a vena contracta effect, is a result of the inability of the fluid to turn the sharp 90 corner.Figure 3.14 Typical flow patterns and contraction coefficientsThe vena contracta effect is a function of the geometry of the outlet. Some typical configurations are shown in Fig. 3.14 along with typical values of the experimentally obtained contraction coefficient, CC=AjAh, where Aj and Ah are the areas of the jet a the vena contracta and the area of the hole, respectively.Other flow meters based on the Bernoulli equation are used to measure flowrates in open channels such as flumes and irrigation ditches. Two of these devices, the sluice gate and the sharp-crested weir, are discussed below under the assumption of steady, inviscid, incompressible flow.Sluice gate geometryWe apply the Bernoulli and continuity equations between points on the free surfaces at (1) and (2) to give:p1+12ρV12+γz1=p2+12ρV22+γz2and Q=V1A1=bV1z1=V2A2=bV2z2With the fact that p1=p2=0:Q=z2b2gz1-z21-z2z12In the limit of z1?z2:Q=z2b2gz1Rectangular, sharp-crested weir geometryFor such devices the flowrate of liquid over the top of the weir plate is dependent on the weir height, Pw, the width of the channel, b, and the head, H, of the water above the top of the weir. Between points (1) and (2) the pressure and gravitational fields cause the fluid to accelerate from velocity V1 to velocity V2. At (1) the pressure is p1=γh, while at (2) the pressure is essentially atmospheric, p2=0. Across the curved streamlines directly above the top of the weir plate (section a–a), the pressure changes from atmospheric on the top surface to some maximum value within the fluid stream and then to atmospheric again at the bottom surface.For now, we will take a very simple approach and assume that the weir flow is similar in many respects to an orifice-type flow with a free streamline. In this instance we would expect the average velocity across the top of the weir to be proportional to 2gH and the flow area for this rectangular weir to be proportional to Hb. Hence, it follows thatQ=C1Hb2gH=C1b2gH323.7 Energy grade line (EGL) and hydraulic grade line (HGL)In this chapter, we neglect losses and/or minor losses, and energy input or output by pumps or turbines:hL=0, hp=0, ht=0p1γ+V122g+z1=p2γ+V222g+z2DefineHGL=pγ+zEGL=pγ+z+V22g point-by point application is graphically displayedHGL corresponds to pressure tap measurement + zEGL corresponds to stagnation tube measurement + zEGL=HGL if V=0hL=fLDV22g i.e., linear variation in L for D, V, and f constant f = friction factor f = fReEGL1=EGL2+hLfor hp=ht=0Pressure tap: p2γ=hStagnation tube: p2γ+V222g=hHelpful hints for drawing HGL and EGL1. EGL=HGL+V22g=HGL for V=02. p=0 ? HGL=z3. for change in D ? change in Vi.e.Change in distance between HGL & EGL and slope change due to change in hLV1A1=V2A2V1πD124=V2πD224V1D12=V2D22? 4. If HGL<z then pγ<0 i.e., cavitation possibleCondition for cavitation:p=pva=2000Nm2Gage pressure pva,g=pva-patm≈-patm=-100,000Nm2pva,gγ≈-10mwhere γ=9810Nm23.9 Limitations of Bernoulli EquationAssumptions used in the derivation Bernoulli Equation:Inviscid Incompressible Steady Conservative body force1) Compressibility Effects: The Bernoulli equation can be modified for compressible flows. A simple, although specialized, case of compressible flow occurs when the temperature of a perfect gas remains constant along the streamline—isothermal flow. Thus, we consider p=ρRT, where T is constant (In general, p, ρ, and T will vary). An equation similar to the Bernoulli equation can be obtained for isentropic flow of a perfect gas. For steady, inviscid, isothermal flow, Bernoulli equation becomesRTdpp+12V2+gz=constThe constant of integration is easily evaluated if z1, p1, and V1 are known at some location on the streamline. The result isV122g+z1+RTglnp1p2=V222g+z22) Unsteady Effects: The Bernoulli equation can be modified for unsteady flows. With the inclusion of the unsteady effect (?V?t≠0) the following is obtained:ρ?V?tds+dp+12ρdV2+γdz=0 (along a streamline)For incompressible flow this can be easily integrated between points (1) and (2) to givep1+12ρV12+γz1=ρs1s2?V?t ds+p2+12ρV22+γz2 (along a streamline)3) Rotational EffectsCare must be used in applying the Bernoulli equation across streamlines. If the flow is “irrotational” (i.e., the fluid particles do not “spin” as they move), it is appropriate to use the Bernoulli equation across streamlines. However, if the flow is “rotational” (fluid particles “spin”), use of the Bernoulli equation is restricted to flow along a streamline.4) Other RestrictionsAnother restriction on the Bernoulli equation is that the flow is inviscid. The Bernoulli equation is actually a first integral of Newton's second law along a streamline. This general integration was possible because, in the absence of viscous effects, the fluid system considered was a conservative system. The total energy of the system remains constant. If viscous effects are important the system is nonconservative and energy losses occur. A more detailed analysis is needed for these cases.The Bernoulli equation is not valid for flows that involve pumps or turbines. The final basic restriction on use of the Bernoulli equation is that there are no mechanical devices (pumps or turbines) in the system between the two points along the streamline for which the equation is applied. These devices represent sources or sinks of energy. Since the Bernoulli equation is actually one form of the energy equation, it must be altered to include pumps or turbines, if these are present. ................
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